R
EPUBLIQUE
A
LGERIENNE
D
EMOCRATIQUE ET
P
OPULAIRE
Minist`ere de l’Enseignement Supérieur et de la Recherche Scientifique
Université de Batna 2
Faculté de Mathématiques et Informatique Département de Mathématiques
Thèse
Pour l’obtention du diplôme de Doctorat Option : ANALYSE ETCONTROLE DESSYSTEMES
EXISTENCE DE LA SOLUTION DE
CERTAINS SYSTEMES
D’EVOLUTION SEMI LINEAIRE
Pr ´esent ´ee par SAMAH GHEGAL
Soutenue le 30/05/2019 devant le jury:
Pr. Salah-Eddine Rebiai Président Université de Batna 2
Dr. Ilhem Hamchi Rapporteur Université de Batna 2
Pr. Farida Lombarkia Examinateur Université de Batna 2
Dr. Maissa Kada Examinateur Université de Batna 2
Dr. Khaled Saoudi Examinateur Université de Khenchla
I owe my gratitude to all those people who have made my journey to this date possible. No matter how many words I write, I can never find words that measure the intensity of my gratitude to all of you. Thank you.
I would like to thank my superviser Dr. Hamchi Ilhem. She always listens, advices, encourages and guides me. She was behind this work and i feel honored to have had a chance to work with her and learn from her.
I am very grateful to the president of the jury, Prof. Rebiai Salah-Eddine and examin-ers: Prof. Lombarkia Farida, Dr. Kada Maissa and Dr. Saoudi Khaled for agreeing to report on this thesis and for their valuable comments and suggestions.
Additionally, I would like to thank Prof . Salim Messaoudi for his willingness to an-swer my numerous questions and especially for his collaboration on one of the results presented in this project.
Of course, this thesis is the culmination of many years of study. I am indebted to all those who guided me on the road of mathematics. I would like to thank my teachers from primary school until now, who have helped me and encouraged me during my studies and especially lovers the mathematics.
I also would like to thank my friends and colleagues, for all the time that we shared to-gether, for their support and care that helped me to stay focused on my graduate study. Most importantly, none of this would have been possible without the love and patience of my family. I want to dedicate this work to my parents Fatima and Ibrahim, my brothers, my husband Abderrezak and my son Mohammed Abdennour.
صخلم فدهلا يسيئرلا نم هذه ةلاسرلا وه ةسارد ةلكشم دوجو رارقتساو لحلا لماشلا ضعبل لا لمج هبش ةيطخ . رت ت زك ةساردلا ىلولأا ىلع ةلداعم ةيدئاز ريغ ةيطخ عم تلاماعم عبنم و حبك ، تابثلإ دوجو لحلا لماشلا هذهل ةلمجلا مدختسن ةقيرط ةعومجملا ةرقتسملا مث نم للاخ قيبطت تاحجارتم كينروموك لصحن ىلع ةجيتن رارقتسلاا . طلستو ةساردلا ةيناثلا ءوضلا ىلع ةلداعم ةجوملا لا ريغ ةيطخ عم و عبنملا تلاماعم حبكلا ةريغتملا ، اضيأ مادختساب قيرط ة ةعومجملا ةرقتسملا و ب قيبطت تاحجارتم كينروموك لصحن ىلع جئاتن دوجو رارقتساو لحلا لماشلا . اريخأ ، سردن ةلداعم ةجوملا لا ريغ لا ةيطخ لماعم يلع يوتحت يتلاو بارطضا نم ةجردلا ىلولأا و ن نيب هنأ امدنع نميهي لماعم حبكلا ( يطخلا جزللاو ) ىلع لماعم بارطضلاا ضفخنت ةقاطلا ةداتعملا نوكيو لحلا .لاماش تاملكلا ةيحاتتفلاا : لماعم ،عبنملا لماعم ،حبكلا لماعم ،بارطضلاا لحلا ،لماشلا رارقتسلاا .
stability of some semi linear evolution systems. The first study focuses on the non-linear hyperbolic equation with damping and source terms. To prove the existence of a global solution of this system, we use some assumptions for initial data. Then, by applying an integral inequality due to Komornik, we obtain the stability result. The second study highlights the nonlinear wave equation with damping and source terms of variable-exponent types. Also, by some assumptions for initial data and Komornik’s integral inequality, we obtained similar results. Finally, we consider the nonlinear wave equation with damping, source and nonlinear first order perturbation terms. We show that when the damping term (linear and viscoelastic) dominates the nonlinear first order perturbation terms, the usual energy is decreases and the solution is global.
keywords: Source term, Dissipative term; Variable exponents; Perturbation term; Global solu-tion; Stability.
CONTENTS
Acknowledgments 2
General introduction 3
1 Global existence and stability of hyperbolic equation with source and
damp-ing terms 9
1.1 Introduction . . . 9
1.2 Preliminary results . . . 10
1.3 Global existence result . . . 11
1.4 Stability result . . . 15
2 Global existence and stability of a nonlinear wave equation with variable exponent nonlinearities 21 2.1 Introduction . . . 21
2.2 Preliminary results . . . 22
2.3 Global existence result . . . 24
2.4 Stability result . . . 27
3 Global existence of the wave equation with nonlinear first order perturba-tion term 34 3.1 Introduction . . . 34
3.3 Global existence result . . . 41
Conclusion 45
Perspectives 46
GENERAL INTRODUCTION
In all this work • T > 0.
• Ω is a bounded domain of Rn(n ∈ N∗), with a smooth boundary ∂Ω.
Literature. During the last few decades, many researchers have been interested in the nonlinear wave equations with source and damping terms with constant exponents:
utt− ∆u + Rt 0g (t − τ ) ∆u (τ ) dτ + ut|ut| m−2 = u |u|p−2, in Ω × (0, T ) , u = 0, on ∂Ω × (0, T ) , u (x, 0) = u0, ut(x, 0) = u1, in Ω, (P) where p, m ≥ 2 are constants and g is a given positive function defined on Ω.
In the absence of the viscoelastic terms (g = 0), system (P) has been largely studied by several authors. Ball [6] showed that in the absence of the damping term ut|ut|
m−2
, the source term u |u|p−2 causes finite-time blow-up of solutions with negative initial energy. Haraux, Zuazua and Kopackova [23], [27] proved that in the absence of the source term, the damping term assures global existence for arbitrary initial data. In the linear damping case m = 2, Levine [29] established a finite-time blow-up result
for negative initial energy. Georgiev and Todorova [18] extended Levine’s result to the nonlinear damping case m > 2. They gave two results:
• if m ≥ p then the global solution exists for arbitrary initial data.
• if p > m then solutions with sufficiently negative initial energy blow-up in finite time.
Messaoudi [30] improved the result of Georgiev and Todorova and proved a finite time blow-up result for solutions with negative initial energy only. Ikehata [24] used some assumptions for initial data, introduced by Sattinger [38] to show that the global solu-tion exists for small enough initial energy. In addisolu-tion, authors in [21], [25] and [37] have addressed this issue.
In the presence of the viscoelastic terms (g 6= 0), Messaoudi in [31] and [32] proved that the solution with negative or positive initial energy blow up in finite time. A result of global existence is also obtained for any initial data. In addition, we refer to [7] , [11], [18], [19], [28] and [42] for other results in this direction.
Recently, many papers have treated the problem with variable exponents. The study of these systems is based on the use of the Lebesgue and Sobolev spaces with vari-able exponents, see for instance [15] and [39]. Antontsev [2] discussed the following equation utt = div
a |∇u|p(x,t)−2∇u+ α∆u + bu |u|σ(x,t)−2+ f, in Ω × (0, T ) ,
u = 0, in ∂Ω × (0, T ) ,
u (x, 0) = u0, ut(x, 0) = u1, in Ω,
where α is a nonnegative constant and a, b, p, σ are given functions. He proved the existence and the blow-up of weak solutions with negative initial energy. Gao [17] studied the following viscoelastic hyperbolic equation
utt− ∆u − ∆utt+ Rt 0 g (t − τ ) ∆u (τ ) dτ + ut|ut| m(.)−2 = u |u|p(.)−2, in Ω × (0, T ) , u = 0, on ∂Ω × (0, T ) , u (x, 0) = u0, ut(x, 0) = u1, in Ω.
General introduction
Using the Galerkin method, the authors proved the existence of weak solutions under suitable assumptions. Another study of Sun et al. [40] where they looked into the following equation
utt− div (a |∇u|) + cut|ut|q(x,t)−1 = bu |u|p(x,t)−1, in Ω × (0, T ) ,
u = 0, on ∂Ω × (0, T ) ,
u (x, 0) = u0, ut(x, 0) = u1, in Ω.
Under specific conditions on the exponents and the initial data, they established a blow-up result. Messaoudi and Talahmeh [34] considered the following equation
utt− div
|∇u|m(.)−2∇u+ µut= u |u|p(.)−2, in Ω × (0, T ) ,
u = 0, on ∂Ω × (0, T ) ,
u (x, 0) = u0, ut(x, 0) = u1, in Ω,
where µ is a nonnegative constant. Under suitable conditions on the exponents, they established a blow-up result for solutions with arbitrary positive initial energy. After that, the same authors in [35] studied the following equation
utt− div
|∇u|r(.)−2∇u+ aut|ut|m(.)−2 = bu |u|p(.)−2, in Ω × (0, T ) ,
u = 0, on ∂Ω × (0, T ) ,
u (x, 0) = u0, ut(x, 0) = u1, in Ω,
where a, b are positive constants and the exponents of nonlinearity m, p, r are given functions. They proved a finite time blow-up result for the solution with negative initial energy as well as for certain solutions with positive initial energy. Also, Messaoudi et al. [36] considered the following equation
utt− ∆u + aut|ut|m(.)−2= bu |u|p(.)−2, in Ω × (0, T ) , u = 0, on ∂Ω × (0, T ) , u (x, 0) = u0, ut(x, 0) = u1, in Ω,
and used the Galerkin method to establish the existence of a unique weak local solution. They also proved that the solutions with negative initial energy blow-up in finite time.
Another kind of system was considered in the literature: System of the wave equa-tion which include linear or nonlinear first order perturbaequa-tion order terms. Cavalcanti and Soriano [14] considered the following hyperbolic boundary problem
∂2u ∂t2 − ∆u + au +P bi ∂u ∂xi = 0, in Ω × (0, ∞) , u = 0, on Γ1× (0, ∞) , ∂u ∂v + β (x) ∂u ∂t = 0 on Γ2× (0, ∞) , u (x, 0) = u0, ∂u∂t (x, 0) = u1, in Ω,
where ∂Ω = Γ1 ∪ Γ2, with restrictions on a, bi and β. They proved the existence,
uniqueness and uniform stability of strong solutions. Cavalcanti et al. [13] studied the following degenerate hyperbolic equations with boundary damping
K (x, t) utt− ∆u + F (x, t, u, ut, ∇u) = f, in Ω × (0, ∞) , u = 0, on Γ1× (0, ∞) , ∂u ∂v + b (x) ut= 0 on Γ2× (0, ∞) , u (x, 0) = u0, ut(x, 0) = u1, in Ω.
Under some assumptions, the authors proved the similar results. Cavalcanti and Guesmia [12] considered the following hyperbolic problem
utt− ∆u + F (x, t, u, ut, ∇u) = f, in Ω × [0, +∞ [, u = 0, on Γ1× [0, +∞ [, u +R0tg (t − s)∂u∂v(s) ds = 0 on Γ2× [0, +∞ [, u (x, 0) = u0, ut(x, 0) = u1, in Ω.
General introduction
They proved that the dissipation given by the memory term is strong enough to assure stability. Guesmia in [20] considered the following semi linear wave internal dissipative term
utt− ∆u + h (∇u) + f (u) + g (ut) = 0, in Ω × [0, +∞ [,
u = 0, on ∂Ω × [0, +∞ [,
u (x, 0) = u0, ut(x, 0) = u1, in Ω,
and the nonlinear boundary feedback
utt− ∆u + h (∇u) + f (u) = 0, in Ω × [0, +∞ [,
u = 0, on Γ1× [0, +∞ [,
∂vu + g (ut) = 0 on Γ2× [0, +∞ [,
u (x, 0) = u0, ut(x, 0) = u1, in Ω,
where f, g : R −→ R and h : Rn −→ R are given continuous nonlinear functions.
Hamchi, in [22], considered the case of linear first order perturbation and improved the result in [20] where she proved an energy decay of solution without any condition on smallness of this term. In her work, she introduced a new geometric multiplier to han-dle the linear first order term and used a suitable nonlinear version of a compactness uniqueness argument in [16] to absorb the lower order terms.
Main contribution. The purpose of this thesis is to study the problem of global ex-istence and stability of solution for some semi linear hyperbolic systems. The first goal considers the study of the global existence and stability of solution for the nonlinear hy-perbolic equation with damping and source terms. The second goal is also centered on the study of these problems for the nonlinear wave equation with damping and source terms of variable-exponent types. The third one is to prove that solution of the wave equation with damping, source and nonlinear first order perturbation terms is global.
Organization of the thesis. This thesis consists of three main chapters in addition to general introduction and conclusion. In the first chapter, we consider the following
hyperbolic equation with source and damping terms utt− div (A∇u) + ut|ut| m−2 = u |u|p−2, inΩ × (0, T ) , u (x, t) = 0, on∂Ω × (0, T ) , u (x, 0) = u0(x) , ut(x, 0) = u1(x) , inΩ,
where p, m ≥ 2. First, we present the theorem of existence of solution. Next, we show that the energy is decreasing and we use some assumptions for initial data to prove a global existence result. Then, by applying an integral inequality due to Komornik, we obtain the stability result. In the second chapter, we consider the following nonlinear wave equation with damping and source terms of variable-exponent types
utt− ∆u + ut|ut| m(.)−2 = u |u|p(.)−2, in Ω × (0, T ) , u = 0, on ∂Ω × (0, T ) , u (x, 0) = u0, ut(x, 0) = u1, in Ω,
where m and p are measurable functions in Ω. First, we present the result of S. Mes-saoudi in [36] concerning the existence of the local solution of this system. Next, we show that the energy is decreasing and by some assumptions for initial data and Ko-mornik’s integral inequality, we obtain the global and stability results. In the third chapter, we consider the following nonlinear wave equation with damping, source and nonlinear first order perturbation terms
utt− ∆u + g ∗ ∆u + a (x) ut+ F (t, ∇u) = |u|p−2u, in Ω × (0, T ) ,
u = 0, on ∂Ω × (0, T ) ,
u (x, 0) = u0, ut(x, 0) = u1, in Ω,
where g, a and F are a functions satisfying some conditions to be specified later. Noting that g ∗ ∆u = R0tg (t − τ ) ∆u (τ ) dτ . First, we show that if the damping terms (linear and viscoelastic) dominated the nonlinear first order perturbation terms then the usual energy is decreasing. After that, we use some assumptions for initial data to prove that the solution of this system is global. Finally, in the conclusion we summarize our results and give some perspectives.
CHAPTER 1
GLOBAL EXISTENCE AND
STABILITY OF HYPERBOLIC
EQUATION WITH SOURCE AND
DAMPING TERMS
1.1
Introduction
The result of this chapter is a case of a result presented in: Congr`es des Mathématiciens Algériens, CMA 2016, Batna, 08-09 Novembre 2016.
In this chapter, we consider the following system
utt− div (A∇u) + ut|ut|m−2 = u |u|p−2, in Ω × (0, T ) ,
u = 0, on ∂Ω × (0, T ) ,
u (x, 0) = u0, ut(x, 0) = u1, in Ω,
(P)
with p, m ≥ 2 and A = (aij(x, t))i,j where, for all i, j = 1, n, aij is a symmetric
such that, for all (x, t) ∈ ¯Ω × [0, +∞[ and ζ ∈ Rn, we have
Aζ.ζ ≥ a0|ζ|2 (1.1)
and
A0ζ.ζ ≤ 0. (1.2)
For the constant case i.e. A = In, many results were obtained concerning the existence,
stability and blow up of solution, see for example [6], [18], [23], [24], [27] and [38]. Concerning the variable coefficient case, the first attempt in this direction was made by Benabderrahmanne and Boukhatem in [8].
Our study have been conducted to analyze the existence of global solution of wave equation with damping and source terms. Then, we employ an integral inequality, due to Komornik [26], to establish a stability result.
1.2
Preliminary results
Regarding the solution of the system (P), we can use the idea of Wu and Tsai in [41] to prove the following theorem.
Theorem 1.1. If u0 ∈ H01(Ω) ∩ H2(Ω) and u1 ∈ H01(Ω) then there exists a unique
strong solutionu in (0, T ) of (P) satisfying u ∈ C ((0, T ) , H1
0(Ω) ∩ H2(Ω)) ,
ut∈ C ((0, T ) , L2(Ω) ∩ L2((0, T ) , H01(Ω)) ,
utt ∈ L2((0, T ) , L2(Ω)) ,
Moreover, the following alternatives hold
(i) T = +∞, or
(ii) T < +∞ and lim
t→T k∇u (t)k 2
2+ kut(t)k 2
2 = +∞.
The decay of the energy of the system (P) is given in the following lemma.
Lemma 1.1. Suppose that (1.2) holds. The energy E of the system (P) is a decreasing function. Here E (t) = 1 2 Z Ω A∇u (t) ∇u (t) dx + 1 2kut(t)k 2 2− 1 p Z Ω |u (t)|pdx.
Chapter 1. Global existence and stability of hyperbolic equation with source and damping terms
Proof. It is enough to multiply the first equation in (P) by ut and integrate over Ω, to
obtain Z Ω utt(t) ut(t) dx− Z Ω div (A∇u (t)) ut(t) dx+ Z Ω |ut(t)| m dx = Z Ω |u (t)|p−2u (t) ut(t) dx.
Then, we use the generalized Green formula and the boundary conditions, to find 1 2 d dt Z Ω |ut(t)| 2 dx + Z Ω A∇u (t) ∇ut(t) dx + Z Ω |ut(t)| m dx = 1 p d dt Z Ω |u (t)|pdx.
This implies that
1 2 d dt R Ω|ut(t)| 2
dx +12dtd RΩA∇u (t) ∇u (t) dx − 12RΩA0∇u (t) ∇u (t) dx +RΩ|ut(t)|mdx = 1pdtd R Ω|u (t)| p dx. So d dtE (t) = 1 2 R ΩA 0 ∇u (t) ∇u (t) dx −R Ω|ut(t)| m dx. (1.3)
Taking into account the hypotheses on A, we find d
dtE (t) ≤ − Z
Ω
|ut(t)|mdx ≤ 0. (1.4)
1.3
Global existence result
To study the problem of global existence of the solution of the system (P), we define the following functions, for all t ∈ (0, T )
I (t) = Z Ω A∇u (t) ∇u (t) dx − Z Ω |u (t)|pdx
and J (t) = 1 2 Z Ω A∇u (t) ∇u (t) dx − 1 p Z Ω |u (t)|pdx.
The proof of the first main result of this chapter requires the following lemma. Lemma 1.2. Suppose that
2 < p ≤ 2n − 1
n − 2, n ≥ 3. Under the assumptions of Lemma 1.1 and supposing that
I (0) > 0 and β := C p ∗ a0 2p a0(p − 2) E (0) p−22 < 1, (1.5)
whereC∗ is the best constant of the embeddingH01(Ω) ,→ Lp(Ω), we have
I (t) > 0 f or all t ∈ (0, T ) .
Proof. By continuity, there exists Tmwhere 0 < Tm < T such that
I (t) ≥ 0, ∀t ∈ (0, Tm) . (1.6)
Now, we will prove that this inequality is strict, we have for all t ∈ (0, Tm)
J (t) = 12RΩA∇u (t) ∇u (t) dx − 1pRΩ|u (t)|pdx
= 12RΩA∇u (t) ∇u (t) dx − 1pRΩA∇u (t) ∇u (t) dx − I (t)
Chapter 1. Global existence and stability of hyperbolic equation with source and damping terms Using (1.6), we obtain Z Ω A∇u (t) ∇u (t) dx ≤ 2p p − 2J (t) , ∀t ∈ (0, Tm) . (1.7) By assumption on A, (1.7) and the definition of E, we find
k∇u (t)k22 ≤ 1 a0 Z Ω A∇u (t) ∇u (t) dx ≤ 2p a0(p − 2) E (t) . (1.8)
Using the nonincreasingness of E, we obtain k∇u (t)k22 ≤ 2p a0(p − 2) E (0) . (1.9) We have R Ω|u (t)| p dx ≤ Cp ∗k∇u (t)k p 2 = C∗pk∇u (t)k p−2 2 k∇u (t)k 2 2. By (1.9), we obtain Z Ω |u (t)|pdx ≤ C∗p 2p a0(p − 2) E (0) p−22 k∇u (t)k22. Using (1.1), we obtain Z Ω |u (t)|pdx ≤ β Z Ω A∇u (t) ∇u (t) dx. (1.10) Since β < 1 then Z Ω |u (t)|pdx < Z Ω A∇u (t) ∇u (t) dx, ∀t ∈ (0, Tm) .
This implies that
Since I (Tm) > 0 and C∗p a0 2p a0(p − 2) E (Tm) p−22 ≤ β < 1.
Then, by repeating the above procedure, we extend Tmto T .
The first main result of this chapter is given in the following theorem. Theorem 1.2. The solution u of (P) is global.
Proof. We have to prove that Z Ω A∇u (t) ∇u (t) dx + kut(t)k 2 2, (1.11) is bounded independently of t.
Based on the definition of E and using (1.7), we get E (t) = J (t) +12kut(t)k22 ≥ p−2 2p R ΩA∇u (t) ∇u (t) dx + 1 2kut(t)k 2 2. (1.12)
Consequently, there exist a constant C > 0 such that Z
Ω
A∇u (t) ∇u (t) dx + kut(t)k 2
2 ≤ CE (t) . (1.13)
By the nonincreasingness of E, we find Z
Ω
A∇u (t) ∇u (t) dx + kut(t)k 2
Chapter 1. Global existence and stability of hyperbolic equation with source and damping terms
1.4
Stability result
To prove the second main result of this chapter, we establish the following lemma. Lemma 1.3. Suppose that
2 < m ≤ 2n
n − 2, n ≥ 3.
Under the assumptions of Lemma 1.2, there exists a positive constantC such that the global solutionu of (P) satisfies
Z
Ω
|u (t)|mdx ≤ CE (t) , f or all t ≥ 0.
Proof. Let c∗ be the best constant of the embedding H01(Ω) ,→ Lm(Ω). Then, we have
R Ω|u (t)| m dx ≤ cm∗ k∇u (t)k m 2 = cm∗ k∇u (t)k m−2 2 k∇u (t)k 2 2. By (1.8), we obtain Z Ω |u (t)|mdx ≤ 2pc m ∗ a0(p − 2) k∇u (t)km−22 E (t) . Then, by (1.9), we obtain R Ω|u (t)| m dx ≤ 2pcm∗ a0(p−2) 2p a0(p−2)E (0) m−22 E (t) .
The second main result of this chapter is given in the following theorem.
Theorem 1.3. Under the assumptions of Lemma 1.3, there exists C, ω > 0 such that the global solution of the system(P ) satisfies
E (t) ≤ CE(0) (1+t)2/(m−2), ∀t ≥ 0 if m > 2. E (t) ≤ CE (0) e−ωt, ∀t ≥ 0 if m = 2.
Proof. Let T > S > 0 and q ≥ 0. Multiplying the first equation of (P) by u (t) Eq(t)
and integrating over Ω × (S, T ), to obtain Z T S Z Ω Eq(t)u (t) utt(t) − u (t) div (A∇u (t)) + u (t) ut(t) |ut(t)|m−2 dxdt = Z T S Z Ω Eq(t) |u (t)|pdxdt.
This implies that Z T S Z Ω Eq(t)(u (t) ut(t))t− |ut(t)| 2 + A∇u (t) ∇u (t) + u (t) ut(t) |ut(t)| m−2 dxdt = Z T S Z Ω Eq(t) |u (t)|pdxdt.
We add, subtract the term Z T S Z Ω Eq(t)βA∇u (t) ∇u (t) + (1 + β) |ut(t)| 2 dxdt
and use (1.10) to get (1 − β)RT S E q(t)R Ω A∇u (t) ∇u (t) + |ut(t)| 2 dxdt +RT S E q(t)R Ω(u (t) ut(t))t− (2 − β) |ut(t)| 2 dxdt +RT S E q(t)R Ωu (t) ut(t) |ut(t)| m−2 dxdt = −RT S E q(t)R Ω[βA∇u (t) ∇u (t) − |u (t)| p ] dxdt ≤ 0.
Chapter 1. Global existence and stability of hyperbolic equation with source and damping terms Then (1 − β)RT S E q(t)R Ω A∇u (t) ∇u (t) + |ut(t)|2− 2p|u (t)|p dxdt ≤ −RT S E q(t)R Ω(u (t) ut(t))tdxdt −RT S E q(t)R Ωu (t) ut(t) |ut(t)| m−2 dxdt + (2 − β)RT S E q(t)R Ω|ut(t)| 2 dxdt. (1.14)
Using the definition of E and the following relation d dt Eq(t) Z Ω u (t) ut(t) dx = qEq−1(t) E0(t) Z Ω u (t) ut(t) dx+Eq(t) Z Ω (u (t) ut(t))tdx,
inequality (1.14) turns into
2 (1 − β)RST Eq+1(t) dt ≤ qRT S E q−1(t) E0 (t)RΩu (t) ut(t) dxdt − RT S d dt E q(t)R Ωu (t) ut(t) dx dt −RT S E q(t)R Ωu (t) ut(t) |ut(t)| m−2 dxdt + (2 − β)RST Eq(t)RΩ|ut(t)| 2 dxdt. (1.15) Let C be a positive generic constant. We estimate the terms in the right of (1.15) as follows qRSTEq−1(t) E0(t)R Ωu (t) ut(t) dxdt ≤ qRT S E q−1(t) −E0(t) 1 2 R Ω|u (t)| 2 dx + 12RΩ|ut(t)| 2dx dt ≤ CRT S E q−1(t) −E0(t) R Ω|∇u (t)| 2 dx +RΩ|ut(t)| 2dx dt ≤ CRT S E q−1(t) −E0(t)hR Ω 1 a0A∇u (t) ∇u (t) dx + R Ω|ut(t)| 2 dxidt.
Based on the definition of E and using (1.7) we get qRST Eq−1(t) E0(t)RΩu (t) ut(t) dxdt ≤ C RT S E q(t) −E0 (t) dt ≤ CEq+1(S) − CEq+1(T ) . That is q Z T S Eq−1(t) E0(t) Z Ω u (t) ut(t) dxdt ≤ CEq(0) E (S) . (1.16)
For the second term, we have −RT S d dt E q(t)R Ωu (t) ut(t) dx dt ≤ Eq(S)RΩu (x, S) ut(x, S) dx − Eq(T ) R Ωu (x, T ) ut(x, T ) dx ≤ Eq(S) R Ωu (x, S) ut(x, S) dx + Eq(T ) R Ωu (x, T ) ut(x, T ) dx ≤ CEq+1(S) + CEq+1(T ) . That is − Z T S d dt Eq(t) Z Ω u (t) ut(t) dx dt ≤ CEq(0) E (S) . (1.17)
For the third term, we use the following Young inequality
XY ≤ r1 Xr1 + 1 r2r2/r1 Yr2, X, Y ≥ 0, > 0 and 1 r1 + 1 r2 = 1. (1.18)
With r1 = m and r2 = m−1m , to find
|u (t)| |ut(t)| m−1
≤ C |u (t)|m+ C|ut(t)| m
Chapter 1. Global existence and stability of hyperbolic equation with source and damping terms
By (1.4) and Lemma 1.3 , we obtain −RT S E q(t)R Ωu (t) ut(t) |ut(t)| m−2 dxdt ≤RT S E q(t)C R Ω|u (t)| m dx + C R Ω|ut(t)| mdx dt ≤ CRT S E q(t)R Ω|u (t)| m dxdt + C RT S E q(t) −E0 (t) dt. That is − Z T S Eq(t) Z Ω u (t) ut(t) |ut(t)| m−2 dxdt ≤ C Z T S Eq+1(t) + CEq(0) E (S) . (1.19) For the last term of (1.15), we have
(2 − β)RST Eq(t)R Ω|ut(t)| 2 dxdt ≤ CRST Eq(t) R Ω|ut(t)| m2/m dxdt.
This implies that
(2 − β) Z T S Eq(t) Z Ω |ut(t)|2dxdt ≤ C Z T S Eq(t)−E0(t)2/mdt. (1.20)
But if we use Young inequality (1.18) with r1 = q+1q and r2 = q + 1, we obtain
Z T S Eq(t)−E0(t) 2/m dt ≤ C Z T S Eq+1(t) dt + 1 C Z T S −E0(t) 2(q+1)/m dt. We take q = m2 − 1 to find Z T S Eq(t)−E0(t)2/mdt ≤ C Z T S Eq+1(t) dt + C Z T S −E0(t)dt.
This implies that Z T S Eq(t)−E0(t) 2/m dt ≤ C Z T S Eq+1(t) dt + CE (S) . (1.21)
Replacing (1.21) in (1.20) to find (2 − β) Z T S Eq(t) Z Ω |ut(t)|2dxdt ≤ C Z T S Eq+1(t) dt + CE (S) . (1.22)
By substituting (1.16), (1.17), (1.19) and (1.22) in (1.15) (with q = m2 − 1), we arrive to
2 (1 − β) Z T S Em2 (t) dt ≤ C Z T S Em2 (t) + CE m 2−1(0) E (S) .
Choosing small enough, to find Z T S Em2 (t) dt ≤ CE m 2−1(0) E (S) . By taking T −→ ∞, we get Z ∞ S Em2 (t) ≤ CE m 2−1(0) E (S) .
CHAPTER 2
GLOBAL EXISTENCE AND
STABILITY OF A NONLINEAR WAVE
EQUATION WITH VARIABLE
EXPONENT NONLINEARITIES
2.1
Introduction
This chapter is the subject of an article written in collaboration with Dr. Hamchi (Uni-versity of Batna 2) and Pr. Messaoudi (Uni(Uni-versity of Dhahran Arabie Saoudite). This article was published in Applicable Analysis Journal.
In this chapter, we consider the following problem utt− ∆u + ut|ut| m(.)−2 = u |u|p(.)−2, in Ω × (0, T ) , u = 0, on ∂Ω × (0, T ) , u (x, 0) = u0, ut(x, 0) = u1, in Ω, (P)
In the case when m, p are constants, there have been many results about the exis-tence and blow-up properties of the solutions, we refer the readers to the bibliography given in [9], [10], [32] and [33]. In the recent years, much attention have been paid to the study of problem with variable exponents. More details on these problems can be found in [1], [2], [3], [4] and [5].
In this chapter, we use some assumptions for initial data to prove the global existence of solution. Then, we employ an integral inequality, due to Komornik [26], to establish a stability result.
2.2
Preliminary results
Let q : Ω −→ [1, ∞] be a measurable function. We define the Lebesgue space with variable exponent q (.) by
Lq(.)(Ω) := {u : Ω −→ R measurable andR
Ω|λu (x)| q(x)
dx < ∞ for some λ > 0}.
Equipped with the following Luxembourg-type norm
kukq(.):= inf{λ > 0 :RΩ u(x) λ q(x) dx ≤ 1} is a Banach space.
To obtain our results, we need the following Lemma [39] Lemma 2.1. If 1 ≤ q1 := ess inf x∈Ωq (x) ≤ q (x) ≤ q2 := ess supx∈Ωq (x) < ∞, then min{kukq1 q(.), kuk q2 q(.)} ≤ R Ω|u (.)| q(x) dx ≤ max{kukq1 q(.), kuk q2 q(.)}, for anyu ∈ Lq(.)(Ω).
Chapter 2. Global existence and stability of a nonlinear wave equation with variable exponent nonlinearities
For the existence of the local solution of problem (P), we refer the reader to [36]. Their result is given in the following theorem.
Theorem 2.1. Suppose that m, p ∈ C ¯Ω with
2 ≤ m1 ≤ m (x) ≤ m2 < n−22n , if n ≥ 3. m (x) ≥ 2, if n = 1, 2, and 2 ≤ p1 ≤ p (x) ≤ p2 < 2n−1n−2, if n ≥ 3, p (x) ≥ 2, if n = 1, 2, where m1 := ess inf x∈Ωm (x) , m2 := ess supx∈Ωm (x) and p1 := ess inf x∈Ωp (x) , p2 := ess supx∈Ωp (x) .
Assume further thatm (.) , p (.) verify the log-H¨older continuity condition |q (x) − q (y)| ≤ − A
log|x−y| for allx, y ∈ Ω,
with |x − y| < δ, A > 0, 0 < δ < 1.
Then, for any(u0, u1) ∈ H01(Ω) × L2(Ω), problem (P) has a unique local solution
u ∈ L∞((0, T ) , H01(Ω)) ,
ut∈ L∞((0, T ) , L2(Ω)) ∩ Lm(.)(Ω × (0, T )) ,
utt ∈ L2((0, T ) , H−1(Ω)) ,
Now, if(u0, u1) ∈ H01(Ω) ∩ H2(Ω) × H01(Ω), problem (P) has a unique strong solution
u ∈ L∞((0, T ) , H1
0(Ω) ∩ H2(Ω)) ,
ut ∈ L∞ (0, T ) , H01(Ω) ∩ Lm(.)(Ω × (0, T )) ,
utt ∈ L2((0, T ) , L2(Ω)) ,
2.3
Global existence result
In order to state and prove our result, we define the following functionals, for all t ∈ (0, T ) I (t) = k∇u (t)k22−R Ω|u (t)| p(x) dx, J (t) = 12k∇u (t)k22−R Ω |u(t)|p(x) p(x) dx, and E (t) = J (t) +12 kut(t)k 2 2.
The proof of the first main result in this chapter requires the following two lemmas. Lemma 2.2. Under the assumptions of Theorem 2.1, E is a decreasing function. Proof. It is enough to multiply the first equation in (P) by ut, integrate it over Ω, to
obtain 1 2 d dt R Ω|ut(t)| 2 dx + 12dtd RΩ|∇u (t)|2dx +RΩ|ut(t)| m(x) dx = dtd RΩ |u(t)|p(x)p(x)dx.
This implies that
d dtE (t) = − R Ω|ut(t)| m(x) dx ≤ 0, ∀t ∈ (0, T ) . (2.1)
Lemma 2.3. Under the assumptions of Theorem 2.1 such that I (0) > 0 and β := max ( Cp1 ∗ 2p1 p1− 2 E (0) p1−22 , Cp2 ∗ 2p1 p1− 2 E (0) p2−22 ) < 1, (2.2)
Chapter 2. Global existence and stability of a nonlinear wave equation with variable exponent nonlinearities
whereC∗ is the best constant of the embeddingH01(Ω) ,→ Lp(.)(Ω) . We have
I (t) > 0 f or all t ∈ (0, T ) .
Proof. By continuity, there exists Tm, where 0 < Tm ≤ T , such that
I (t) ≥ 0, ∀t ∈ (0, Tm) . (2.3)
Now, we will prove that this inequality is strict. We have for all t ∈ (0, T ) J (t) = 12k∇u (t)k22−R Ω |u(t)|p(x) p(x) dx ≥ 1 2k∇u (t)k 2 2− 1 p1 k∇u (t)k 2 2 − I (t) ≥ p1−2 2p1 k∇u (t)k 2 2+ 1 p1I (t) . Using (2.3), we obtain k∇u (t)k22 ≤ 2p1 p1 − 2 J (t) , ∀t ∈ (0, Tm) . (2.4)
By the definition of E, we find
k∇u (t)k22 ≤ 2p1 p1− 2
E (t) . (2.5)
Using the nonincreasingness of E, we obtain k∇u (t)k22 ≤ 2p1
p1− 2
E (0) , ∀t ∈ (0, Tm) . (2.6)
On the other hand, by Lemma 2.1, we have Z Ω |u (t)|p(x)dx ≤ maxnku (t)kp1 p(.), ku (t)k p2 p(.) o .
Hence, we have R Ω|u (t)| p(x) dx ≤ max {Cp1 ∗ k∇u (t)k p1 2 , C p2 ∗ k∇u (t)k p2 2 } ≤ maxCp1 ∗ k∇u (t)k p1−2 2 , C p2 ∗ k∇u (t)k p2−2 2 k∇u (t)k 2 2. By (2.6), we obtain Z Ω |u (t)|p(x)dx ≤ β k∇u (t)k22, ∀t ∈ (0, Tm) . (2.7) Since β < 1 then Z Ω |u (t)|p(x)dx < k∇u (t)k22, ∀t ∈ (0, Tm) .
This implies that
I (t) > 0, ∀t ∈ (0, Tm) , hence I (Tm) > 0 and max ( Cp1 ∗ 2p1 p1− 2 E (Tm) p1−22 , Cp2 ∗ 2p1 p1− 2 E (Tm) p2−22 ) ≤ β < 1.
Then, by repeating the above procedure, we extend Tmto T .
The first main result in this chapter is given in the following theorem.
Theorem 2.2. Under the assumptions of Lemma 2.3, the local solution u of (P) is global.
Chapter 2. Global existence and stability of a nonlinear wave equation with variable exponent nonlinearities
Proof. Based on the definition of E and using (2.4) we get E (t) = J (t) +12kut(t)k 2 2 ≥ p1−2 2p1 k∇u (t)k 2 2+ 1 2kut(t)k 2 2.
Consequently, a constant C > 0 exists such that k∇u (t)k22+ kut(t)k 2
2 ≤ CE (t) . (2.8)
By using the nonincreasingness of E, we find k∇u (t)k22+ kut(t)k
2
2 ≤ CE (0) .
This implies that the local solution u of (P) is global and bounded.
2.4
Stability result
To prove the second main result in this chapter, we establish the following lemma. Lemma 2.4. Under the assumptions of Lemma 2.3, there exists a positive constant C, such that the global solutionu of (P) satisfies
Z
Ω
|u (t)|m(x)dx ≤ CE (t) , f or all t ≥ 0.
Proof. Let c∗be the best constant of the embedding H01(Ω) ,→ Lm(.)(Ω); then, we have
R Ω|u (t)| m(x) dx ≤ maxnku (t)km1 m(.), ku (t)k m2 m(.) o ≤ max {cm1 ∗ k∇u (t)k m1 2 , c m2 ∗ k∇u (t)k m2 2 } ≤ maxcm1 ∗ k∇u (t)k m1−2 2 , c m2 ∗ k∇u (t)k m2−2 2 k∇u (t)k 2 2.
By (2.5), we obtain the desired result Z Ω |u (t)|m(x)dx ≤ maxcm1 ∗ k∇u (t)k m1−2 2 , c m2 ∗ k∇u (t)k m2−2 2 2p1 p1− 2 E (t) .
The following theorem gives the stability of the global solution u.
Theorem 2.3. Under the assumptions of Lemma 2.3, there exists constants C, ω > 0 such that the global solution of (P) satisfies
E (t) ≤ CE(0) (1+t)2/(m2−2), ∀t ≥ 0 if m2 > 2. E (t) ≤ CE (0) e−ωt, ∀t ≥ 0 if m2 = 2.
Proof. Let T > S > 0 and q ≥ 0. Multiplying (P) by u (t) Eq(t) and integrating over
Ω × (S, T ), we obtain Z T S Z Ω Eq(t) h u (t) utt(t) − u (t) ∆u (t) + u (t) ut(t) |ut(t)|m(x)−2 i dxdt = Z T S Z Ω Eq|u (t)|p(x)dxdt.
This implies that Z T S Z Ω Eq(t) h (u (t) ut(t))t− |ut(t)| 2 + |∇u (t)|2+ u (t) ut(t) |ut(t)| m(x)−2i dxdt = Z T S Z Ω Eq(t) |u (t)|p(x)dxdt.
We add and subtract the term Z T S Z Ω Eq(t)β |∇u (t)|2+ (1 + β) |ut(t)| 2 dxdt
and use (2.7) to get
(1 − β)RST Eq(t)R Ω |∇u (t)| 2 + |ut(t)| 2 dxdt +RST Eq(t)R Ω(u (t) ut(t))t− (2 − β) |ut(t)| 2 dxdt +RST Eq(t)R Ωu (t) ut(t) |ut(t)| m(x)−2 dxdt = −RT S E q(t)R Ω h β |∇u (t)|2− |u (t)|p(x)idxdt ≤ 0.
Chapter 2. Global existence and stability of a nonlinear wave equation with variable exponent nonlinearities Then (1 − β)RST Eq(t)R Ω |∇u (t)|2 + |ut(t)|2 − 2 |u(t)|p(x) p(x) dxdt ≤ −RT S E q(t)R Ω(u (t) ut(t))tdxdt + (2 − β)RST Eq(t)R Ω|ut(t)| 2 dxdt −RT S E q(t)R Ωu (t) ut(t) |ut(t)| m(x)−2 dxdt. (2.9)
Using the definition of E and the following relation d dt Eq(t) Z Ω u (t) ut(t) dx = qEq−1(t) E0(t) Z Ω u (t) ut(t) dx+Eq(t) Z Ω (u (t) ut(t))tdx,
inequality (2.9) turns into
2 (1 − β)RST Eq+1(t) dt ≤ qRT S E q−1(t) E0(t)R Ωu (t) ut(t) dxdt − RT S d dt E q(t)R Ωu (t) ut(t) dx dt −RT S E q(t)R Ωu (t) ut(t) |ut(t)| m(x)−2 dxdt + (2 − β)RST Eq(t)R Ω|ut(t)| 2 dxdt. (2.10) Let C be a positive generic constant. We estimate the terms in the right-hand side of (2.10) as follows qRST Eq−1(t) E0(t)R Ωu (t) ut(t) dxdt ≤ qRT S E q−1(t) −E0(t) 1 2 R Ω|u (t)| 2 dx +12RΩ|ut(t)| 2dx dt ≤ CRT S E q−1(t) −E0(t) R Ω|∇u (t)| 2 dx +RΩ|ut(t)| 2dx dt.
Thus, by (2.8), we find qRST Eq−1(t) E0(t)RΩu (t) ut(t) dxdt ≤ C RT S E q(t) −E0 (t) dt ≤ CEq+1(S) − CEq+1(T ) ≤ CEq(0) E (S) . (2.11)
For the second term, we have
−RT S d dt E q(t)R Ωu (t) ut(t) dx dt ≤ Eq(S)R Ωu (x, S) ut(x, S) dx − E q(T )R Ωu (x, T ) ut(x, T ) dx ≤ Eq(S) R Ωu (x, S) ut(x, S) dx + Eq(T ) R Ωu (x, T ) ut(x, T ) dx ≤ CEq+1(S) + CEq+1(T ) ≤ CEq(0) E (S) . (2.12) For the third term, we use the following Young inequality
XY ≤ r1 Xr1 + 1 r2r2/r1 Yr2, X, Y ≥ 0, > 0 and 1 r1 + 1 r2 = 1.
With r1(x) = m (x), r2(x) = m(x)−1m(x) and < 1, we find
Chapter 2. Global existence and stability of a nonlinear wave equation with variable exponent nonlinearities
By (2.1) and Lemma 2.4 , we have
−RT S E q(t)R Ωu (t) ut(t) |ut(t)| m(x)−2 dxdt ≤RT S E q(t)hCR Ω|u (t)| m(x) dx + C R Ω|ut(t)| m(x) dxidt ≤ CRT S E q(t)R Ω|u (t)| m(x) dxdt + C RT S E q(t) −E0(t) dt ≤ CRT S E q+1(t) + C Eq(0) E (S) . (2.13)
For the last term of (2.10), we have (2 − β)RST Eq(t)RΩ|ut(t)| 2 dxdt = (2 − β)RST Eq(t)hR Ω−|ut(t)| 2 dx +RΩ +|ut(t)| 2 dxidt ≤ CRT S E q(t) R Ω−|ut(t)| m22/m2 dx +RΩ +|ut(t)| m12/m1 dx dt ≤ CRT S E q(t) R Ω|ut(t)| m(x)2/m2 dx +RΩ|ut(t)| m(x)2/m1 dx dt. This implies (2 − β)RST Eq(t)R Ω|ut(t)| 2 dxdt ≤ CRST Eq(t) −E0(t)2/m2 dt +CRST Eq(t) −E0(t)2/m1 dt. (2.14)
We use Young’s inequality with r1 = q+1q and r2 = q + 1, we have
Z T S Eq(t)−E0(t)2/m2dt ≤ q q + 1 Z T S Eq+1(t) dt+ 1 (q + 1) q Z T S −E0(t)2(q+1)/m2dt. We take q = m2 2 − 1 to find Z T S Eq(t)−E0(t) 2/m2 dt ≤ C Z T S Eq+1(t) dt + C Z T S −E0(t)dt.
This implies Z T S Eq(t)−E0(t)2/m2dt ≤ C Z T S Eq+1(t) dt + CE (S) . (2.15)
On the other hand, we have Z T S Eq(t)−E0(t)2/m1dt ≤ C Z T S Eq+1(t) dt + CE (S) . (2.16) Indeed, • if m1 = 2 then Z T S Eq(t)−E0(t) 2/m1 dt ≤ CE (S) ≤ C Z T S Eq+1(t) dt + CE (S) .
• if m1 > 2, we use Young’s inequality with r1 = mm1−21 and r2 = m21, to obtain
RT S E q(t) −E0 (t)2/m1 ≤ CRT S E qm1/(m1−2)(t) dt + C RT S −E 0 (t) dt ≤ CRT S E qm1/(m1−2)(t) dt + C E (S) . We notice that qm1 m1−2 = q + 1 + m2−m1 m1−2 , then RT S E q(t) −E0 (t)2/m1 ≤ C (E (S))m2−m1m1−2 RT S E q+1(t) dt + C E (S) ≤ CRT S E q+1(t) dt + C E (S) .
We insert (2.15) and (2.16) in (2.14) to obtain
(2 − β) Z T S Eq(t) Z Ω |ut(t)|2dxdt ≤ C Z T S Eq+1(t) dt + CE (S) . (2.17)
By substituting (2.11), (2.12), (2.13) and (2.17) in (2.10) (with q = m2
2 − 1), we arrive to 2 (1 − β) Z T S Em22 (t) dt ≤ C Z T S Em22 (t) + CEm22 −1(0) E (S) .
Chapter 2. Global existence and stability of a nonlinear wave equation with variable exponent nonlinearities
Choosing < 1 small enough Z T S Em22 (t) dt ≤ CEm22 −1(0) E (S) . By taking T −→ ∞, we get Z ∞ S Em22 (t) ≤ CE m2 2 −1(0) E (S) .
GLOBAL EXISTENCE OF THE WAVE
EQUATION WITH NONLINEAR
FIRST ORDER PERTURBATION
TERM
3.1
Introduction
This chapter is the subject of an article written in collaboration with Dr. Hamchi. This article was submitted.
In this chapter, we study the system
utt− ∆u + g ∗ ∆u + a (x) ut+ F (t, ∇u) = |u| p−2 u, in Ω × (0, T ) , u = 0, on ∂Ω × (0, T ) , u (x, 0) = u0, ut(x, 0) = u1, in Ω, (P) where g ∗ υ = Z t 0 g (t − τ ) υ (τ ) dτ f or all t ≥ 0.
Chapter 3. Global existence of the wave equation with nonlinear first order perturbation term
Under the following assumptions
(H1) Assumption on the viscoelastic damping term g is a C1(R+) positive
decreas-ing function satisfydecreas-ing
1 − Z ∞
0
g (s) ds = l > 0.
(H2) Assumption on the linear damping term
0 < a1 := inf
x∈Ωa (x) ≤ a (x) ≤ a2 := supx∈Ωa (x) < ∞, ∀x ∈ Ω.
(H3) Assumption on the nonlinear perturbation term F is a C1(R+× Rn) function
satisfying
|F (t, U )| ≤p2a1 g (t) |U | , ∀t ≥ 0, ∀ U ∈ Rn.
(H4) Assumption on the source term
p > 2 if n = 1, 2 and
2 ≤ p < 2(n−1)n−2 if n ≥ 3.
In the absence of the nonlinear first order perturbation term, the global existence and nonexistence problems of the wave equation with polynomial source and damping terms have been largely studied by several authors, see for instance [7], [11], [18],[28], [30], [31] and [32].
Concerning the wave equation which include linear or nonlinear first perturbation order terms, an uniform decay rate estimates was obtained under strong hypothesis on these terms, see [12], [14] and [16]. Hamchi, in [22], considered the case of linear first order perturbation and improved the result in [20] where she has proved an energy decay of solution without any condition on smallness of this term.
In this chapter, we show that if the damping terms dominated the nonlinear first order perturbation terms then the usual energy is decreasing. After that, by some assumptions for initial data we prove our main result.
3.2
Preliminary results
As in [7] and [41], the existence result of (P ) can be established by Galerkin method combined with the well known contraction mapping theorem. This result is given in the following theorem.
Theorem 3.1. If u0 ∈ H01(Ω) ∩ H2(Ω) and u1 ∈ H01(Ω) then there exists a unique
maximal solutionu in (0, T ) of (P ) satisfying
u ∈ C ((0, T ) , H01(Ω) ∩ H2(Ω)) ,
ut∈ C ((0, T ) , L2(Ω) ∩ L2((0, T ) , H01(Ω)) ,
utt ∈ L2((0, T ) , L2(Ω)) ,
Moreover, the following alternatives hold
(i) T = +∞, or
(ii) T < +∞ and lim
t→T k∇u (t)k 2
2+ kut(t)k 2
2 = +∞.
Consider the usual energy functional for the solution u of the system (P ) defined for all t ∈ (0, T ) by E (t) = 12kut(t)k22+ 12 1 −R0tg (τ ) dτk∇u (t)k22+ 12(g ◦ ∇u) (t) −1pku (t)kpp, where (g ◦ v) (t) = Z t 0 g (t − τ ) kv (t) − v (τ )k22dτ . Lemma 3.1. E is a decreasing function.
Proof. If we multiply the first equation in (P) by ut(t), integrate it over Ω, we obtain
R
Ωa (x) |ut(t)| 2
Chapter 3. Global existence of the wave equation with nonlinear first order perturbation term where I1 = − R Ωutt(t) ut(t) dx, I2 = R Ω∆u (t) ut(t) dx, I3 = R Ω|u (t)| p−2 u (t) ut(t) dx, and I4 = − R Ω(g ∗ ∆u) (t) ut(t) dx. We have I1 = − 1 2 d dtkut(t)k 2 2. (3.2)
By the Green formula and the bounded conditions, we obtain I2 = − R Ω∇u (t) ∇ut(t) dx = − 1 2 d dtk∇u (t)k 2 2, (3.3) and I3 = 1 p d dtku (t)k p p. (3.4)
For I4, we use the Green formula and the bounded conditions, to find
I4 = − R Ω Rt 0 g (t − τ ) ∆u (τ ) ut(t) dτ dx = Rt 0 g (t − τ ) R Ω∇u (τ ) ∇ut(t) dx dτ.
We add and subtract the term Z t 0 g (t − τ ) Z Ω ∇u (t) ∇ut(t) dx dτ,
to get
I4 = −
Rt
0g (t − τ )
R
Ω(∇u (t) − ∇u (τ )) ∇ut(t) dx dτ
+R0tg (t − τ )RΩ∇u (t) ∇ut(t) dx dτ
= −12R0tg (t − τ )dtd RΩ|∇u (t) − ∇u (τ )|2dxdτ +12R0tg (t − τ )dtd RΩ|∇u (t)|2dxdτ. By a variable change, we find
I4 = −12 Rt 0g (t − τ ) d dtk∇u (t) − ∇u (τ )k 2 2dτ + 1 2 Rt 0 g (τ ) d dtk∇u (t)k 2 2dτ. (3.5)
If we apply the following Leibniz integral rule
d dt RB(t) A(t) f (t, τ ) dτ =RA(t)B(t)∂f (t,τ )∂t dτ + f (t, B (t))dtdB (t) − f (t, A (t))dtdA (t) , (3.6) with
A (t) = 0, B (t) = t and f (t, τ ) = −12g (t − τ ) k∇u (t) − ∇u (τ )k22,
we obtain −1 2 d dt(g ◦ ∇u) (t) = − 1 2 d dt Rt 0 g (t − τ ) k∇u (t) − ∇u (τ )k 2 2dτ
= −21 g0 ◦ ∇u (t) − 12R0tg (t − τ )dtd k∇u (t) − ∇u (τ )k22dτ. This implies that
−1 2 Rt 0 g (t − τ ) d dtk∇u (t) − ∇u (τ )k 2 2dτ = − 1 2 d dt(g ◦ ∇u) (t) + 1 2 g 0 ◦ ∇u (t) . (3.7)
Chapter 3. Global existence of the wave equation with nonlinear first order perturbation term
Now, if we apply (3.6) with
A (t) = 0, B (t) = t and f (t, τ ) = 1 2g (τ ) k∇u (t)k 2 2, we obtain 1 2 d dt Rt 0 g (τ ) k∇u (t)k 2 2dτ = 1 2 Rt 0 d dtg (τ ) k∇u (t)k 2 2 dτ + 1 2g (t) k∇u (t)k 2 2 = 1 2 Rt 0 g (τ ) d dtk∇u (t)k 2 2dτ + 1 2g (t) k∇u (t)k 2 2.
This implies that
1 2 Rt 0g (τ ) d dtk∇u (t)k 2 2dτ = 1 2 d dt Rt 0 g (τ ) k∇u (t)k 2 2dτ − 1 2g (t) k∇u (t)k 2 2. (3.8)
We insert (3.7) and (3.8) in (3.5) to obtain
I4 = −12dtd (g ◦ ∇u) (t) + 12 g 0 ◦ ∇u (t) + 1 2 d dt Rt 0 g (τ ) k∇u (t)k 2 2dτ −1 2g (t) k∇u (t)k 2 2. (3.9)
By substituting (3.2), (3.3), (3.4) and (3.9) in (3.1) and using (H2), we obtain
a1kut(t)k 2 2+ R ΩF (t, ∇u (t)) ut(t) dx ≤ − 1 2 d dtkut(t)k 2 2− 1 2 d dt(g ◦ ∇u) (t) −1 2 d dt h 1 −Rt 0g (τ ) dτ k∇u (t)k22i +12 g0◦ ∇u (t) − 12g (t) k∇u (t)k22 +1pdtd ku (t)kpp.
Using the definition of E, we arrive to d dtE (t) = d dt h 1 2kut(t)k 2 2+ 1 2 1 −R0tg (τ ) dτk∇u (t)k22+ 12(g ◦ ∇u) (t) −1pku (t)kppi ≤ 1 2 g 0 ◦ ∇u (t) − 1 2g (t) k∇u (t)k 2 2− a1kut(t)k 2 2− R ΩF (t, ∇u (t)) ut(t) dx. Since g0 ≤ 0 then d dtE (t) ≤ − 1 2g (t) k∇u (t)k 2 2− a1kut(t)k 2 2+ Z Ω |F (t, ∇u (t))| |ut(t)| dx.
We use the following Young inequality
XY ≤ 2X 2 + 1 2Y 2 , f or all X, Y ≥ 0 and > 0, with X = |F (t, ∇u (t))| and Y = |ut(t)| , to find d dtE (t) ≤ 1 2 Z Ω |F (t, ∇u (t))|2 − g (t) |∇u (t)|2 + 1 2− a1 kut(t)k 2 2. We take = 2a1 1 to obtain d dtE (t) ≤ 1 2 Z Ω 1 2a1 |F (t, ∇u (t))|2− g (t) |∇u (t)|2 . Thus, by (H3), we find d dtE (t) ≤ 0, ∀ t ∈ (0, T ) .
Chapter 3. Global existence of the wave equation with nonlinear first order perturbation term
3.3
Global existence result
To study the global existence of the solution of the system (P ), we define the following functionals for all t ∈ (0, T )
I (t) = 1 −R0tg (τ ) dτ k∇u (t)k22 + (g ◦ ∇u) (t) − ku (t)kpp, and J (t) = 121 −Rt 0g (τ ) dτ k∇u (t)k22+ 12(g ◦ ∇u) (t) −1pku (t)kpp.
The proof of our main result in this chapter requires the following lemma. Lemma 3.2. If I (0) > 0, and β := C p ∗ l 2p l (p − 2)E (0) p−22 < 1, (3.10)
whereC∗ is the best constant of the embedding
H01(Ω) ,→ Lp(Ω) .
Then
I (t) > 0 f or all t ∈ (0, T ) .
Proof. By continuity, there exists Tmwhere 0 < Tm ≤ T such that
Using the definition of I and J , we obtain p−2 2p 1 −R0tg (τ ) dτk∇u (t)k22 = J (t) − 1pI (t) − p−22p (g ◦ ∇u) (t) . By (3.11) and (H1), we get 1 − Z t 0 g (τ ) dτ k∇u (t)k22 ≤ 2p p − 2J (t) . But l k∇u (t)k22 ≤ 1 − Z t 0 g (τ ) dτ k∇u (t)k22. (3.12) Then l k∇u (t)k22 ≤ 2p p − 2J (t) . (3.13) Noting that J (t) = E (t) − 1 2kut(t)k 2 2 ≤ E (t) . Then, we have l k∇u (t)k22 ≤ 2p p − 2E (t) . By the nonincreasingness of E, we find
k∇u (t)k22 ≤ 2p
l (p − 2)E (0) . (3.14)
On the other hand
ku (t)kpp ≤ Cp ∗ k∇u (t)k p 2 = C∗p l k∇u (t)k p−2 2 l k∇u (t)k 2 2.
Chapter 3. Global existence of the wave equation with nonlinear first order perturbation term Then, by (3.12), we get ku (t)kpp ≤ C p ∗ l k∇u (t)k p−2 2 1 − Z t 0 g (τ ) dτ k∇u (t)k22. By (3.14), we have ku (t)kpp ≤ C p ∗ l 2p l (p − 2)E (0) p−22 1 − Z t 0 g (τ ) dτ k∇u (t)k22.
This implies that
ku (t)kpp ≤ β 1 − Z t 0 g (τ ) dτ k∇u (t)k22. Since β < 1 then ku (t)kpp < 1 − Z t 0 g (τ ) dτ k∇u (t)k22, ∀t ∈ (0, Tm) . Hence I (t) =1 −R0tg (τ ) dτk∇u (t)k22+ (g ◦ ∇u) (t) − ku (t)kpp ≥1 −R0tg (τ ) dτk∇u (t)k22− ku (t)kpp > 0. That is I (t) > 0, ∀t ∈ (0, Tm) .
Finally, we can see that
I (Tm) > 0 and Cp ∗ l 2p l (p − 2)E (Tm) p−22 ≤ β.
Then, by repeating the above procedure, we extend Tmto T .
The main result of this chapter is given in the following theorem. Theorem 3.2. The solution u of (P) is bounded and global.
Proof. Based on (3.13), we get
(p−2)l 2p k∇u (t)k 2 2+ 1 2kut(t)k 2 2 ≤ J (t) + 1 2kut(t)k 2 2 = E (t) .
Consequently, it exists a constant C > 0 such that
k∇u (t)k22+ kut(t)k22 ≤ CE (t) .
By the nonincreasingness of E, we find
k∇u (t)k22+ kut(t)k 2
2 ≤ CE (0) .
This implies that u is bounded.
CONCLUSION
In this thesis, we have studied the existence and the stability of global solution of some semi linear wave systems.
First, we have studied global existence of hyperbolic equations with damping and source terms using some assumptions for initial data. Then, by applying an integral inequality due to Komornik, we have obtained the stability result.
Many researches have been interested by nonlinear wave equations with variable-exponent, on which studies focus on blow up of solution. However, the global solution have not been studied for this type of equations. This issue have been the subject of a detailed study. Our results have been proved using some assumptions for initial data. Then, by applying an integral inequality due to Komornik, we have obtained the stabil-ity result.
In the literature, we have observed that the wave equations having damping and nonlinear first order perturbation terms have been largely studied. Thus, these equations may be considered with a source term. Actually, we have proved the global existence of solutions of the nonlinear wave equation with damping, source, and nonlinear first order perturbation terms.
As a perspective, it would be interesting to study the existence of the global solution of the system: utt− div |∇u|r(.)−2∇u+ aut|ut| m(.)−2 = bu |u|p(.)−2.
Given the fact that Messaoudi and Talahmeh [35] have proved a finite time blow-up result for the solution.
In the same context, the stability of wave equation with variable exponent in the case when the source term dominates the damping term also can be explored.
As an extension of the system studied in the chapter 2, on which we have considered the variables exponents as p (x) and m (x),the case of time varying exponents:
utt− ∆u + ut|ut|
m(x,t)−2
= u |u|p(x,t)−2.
can be investigated. Actually, these type of systems were studied by several authors. For instance, Antontsev [2] and Sun et al. [40] have studied the existence and blow-up of the solution for these systems.
In the chapter 3 we have studied the global existence of the solution of the wave equation with nonlinear source, damping and first order perturbation term. Subsequently, the stability of this system is another challenge.
BIBLIOGRAPHY
[1] R. Aboulaich and D. Meskine. New diffusion models in image processing. Com-put. Math. Appl, 56:874 – 882, 2008.
[2] S. Antontsev. Wave equation with p(x, t)-laplacian and damping term: Existence and blow-up. J. Difference Equ. Appl, 3:503 – 525, 2011.
[3] S. Antontsev and S. Shmarev. Elliptic equations with anisotropic nonlinearity and nonstandrd growth condition. HandBook of Differential Equations. Station-ary Partial Differential Equations. Elsevier/North Holland. Amsterdam, 3:1 – 100, 2006.
[4] S. Antontsev and S. Shmarev. Blow-up of solutions to parabolic equations with nonstandard growth conditions. J. Comput. Appl. Math, 234:2633 – 2645, 2010. [5] S. Antontsev and V. Zhikov. Higher integrability for parabolic equation of p(x,
t)-laplacian type. Adv. Differential Equation, 10:1053 – 1080, 2005.
[6] J. Ball. Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. The Quarterly Journal of Mathematics, 28:473 – 486, 1977.
[7] S. Berrimi and S. Messaoudi. Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal, 64:2314 – 2331, 2006.
[8] Y. Boukhatem and B. Benabderrahmanne. Blow up of solutions for a semilinear hyperbolic equation. J. Qual. Theory Differ. Eqns, 40:1 – 12, 2012.
[9] M. Cacalcanti and H. Oquendo. Frictional versus viscoelastic damping in a semi-linear wave equation. SIAM J. Control Optim., 42:1310 – 1324, 2003.
[10] M. Cavalcanti, V. Cavalcanti, and J. Ferreira. Existence and uniform decay for nonlinear viscoelastic equation with strong damping. Math.Appl.Sci, 24:1043 – 1053, 2001.
[11] M. Cavalcanti, V. Cavalcanti, and J. Soriano. Exponential decay for the solution of semilinear viscoelastic wave equation with localized damping. Elect. J. Diffe. Eqns, 44:1 – 14, 2002.
[12] M. Cavalcanti and A. Guesmia. General decay rates of solutions to a nonlinear wave equation with boundary condition of memory type. Differential and Integral Equations, 18:583 – 600, 2005.
[13] M. Cavalcanti, N. Lar’kin, and J. Soriano. On solvability and stability of solutions of nonlinear degenerate hyperbolic equations with boundary damping. Funkcialaj Ekvacioj· January, 41:271 – 289, 1998.
[14] M. Cavalcanti and J. Soriano. On solvability and stability of the wave equation with lower order terms and boundary damping. Elect. J. Diffe. Eqns, 18:61 – 78, 1997.
[15] L. Diening, P. Hasto, P. Harjulehto, and M. Ruzicka. Lebesgue and sobolev spaces with variable exponents. Springer-Verlag, Heidelberg, 2011.
[16] S. Feng and D. Feng. Nonlinear internal damping of wave equations with variable coefficients. Acta Math. Sin. – Engl. Ser, 20:1057 – 1072, 2004.
[17] Y. Gao and W. Gao. Existence of weak solutions for viscoelastic hyperbolic equa-tions with variable exponents. Boundary Value Problems, 2013:1 – 8, 20013. [18] V. Georgiev and G. Todorova. Existence of a solution of the wave equation with
nonlinear damping and source terms. Journal of Differential Equations, 109:295 – 308, 1994.
[19] S. Ghegal, I. Hamchi, and S. Messaoudi. Global existence and stability of a non-linear wave equation with variable-exponent nonnon-linearities. Applicable Analysis, 2018.
Bibliography
[20] A. Guesmia. A new approach of stabilization of nondissipative distributed sys-tems. SIAM. J. Control. Optim, 42:24 – 52, 2003.
[21] T. Ha. Blow-up of semilinear wave equation with boundary damping and source terms. J. Math. Anal. Appl, 390:328 – 334, 2012.
[22] I. Hamchi. Uniform decay rates for second order hyperbolic equations with vari-able coefficients. Assmptptic Analysis journal, 57:71 – 82, 2008.
[23] A. Haraux and E. Zuazua. Decay estimates for some semilinear damped hyper-bolic problems. Arch. Rational Mech, 150:191 – 206, 1988.
[24] R. Ikehata. Some remarks on the wave equation with nonlinear damping and source term. Nonlinear analysis. theory. methods and application, 27:1165 – 1175, 1995.
[25] H. Jianghao and L. Cai. Global existence and blowup of solutions for nonlin-ear coupled wave equations with viscoelastic term. Mathematical Methods in the Applied Sciences, 39:1977 – 1989, 2016.
[26] V. Komornik. Exact controllability and stabilization. The Multiplier Method. Masson-John Wiley Paris, 1994.
[27] M. Kopackova. Remarks on bounded solutions of a semilinear dissipative hyper-bolic equation. Comment. Math. Univ. Carolin, 30:713 – 719, 1989.
[28] H. Levine. Instability and nonexistence of global solutions of nonlinear wave equation of the putt = au + f (u). Trans. Amer. Math. Soc, 192:1 – 21, 1974.
[29] H. Levine. Some additional remarks on the nonexistence of global solutions to nonlinear wave equation. SIAM J. Math. Anal, 5:138 – 146, 1974.
[30] S. Messaoudi. Blow up in a nonlinearly damped wave equation. Mathematische Nachrichten, 231:1 – 7, 2001.
[31] S. Messaoudi. Blow up and global existence in a nonlinear viscoelastic wave equation. Mathematische Nachrichten, 260:58 – 66, 2003.
[32] S. Messaoudi. Blow up of positive initial energy solutions of a nonlinear viscoelas-tic hyperbolic equation. J. Math. Anal. Appl, 320:902 – 915, 2006.
[33] S. Messaoudi. General decay of solutions of a viscoelastic equation. J. Math. Anal. Appl, 341:1457 – 1467, 2008.
[34] S. Messaoudi and A. Talahmeh. A blow-up result for a nonlinear wave equation with variable-exponent nonlinearities. Applicable Analysis, 96:1509 – 1515, 2017. [35] S. Messaoudi and A. Talahmeh. Blowup in solutions of a quasilinear wave equa-tion with variable-exponent nonlinearities. Mathematical Methods in the Applied Sciences, 40:1099 – 1476, 2017.
[36] S. Messaoudi, A. Talahmeh, and H. Jamal. Nonlinear damped wave equation: Existence and blow-up. Computers and Mathematics with applications, 74:3024– 3041, 2017.
[37] J. Park and T. Ha. Existence and asymptotic stability for the semilinear wave equation with boundary damping and source term. J. Math. Phys, 49, 2008. [38] D. Sattinger. On global solution of nonlinear hyperbolic equation. Arch ration.
Mech. Analysis, 30:148 – 172, 1968.
[39] A. Stanislav and S. Sergey. Evolution pdes with nonstandard growth condi-tions: Existence, uniqueness. localization. blow-up. Atlantis Studies in Differential Equations, 4:1 – 417, 2015.
[40] L. Sun, Y. Ren, and W. Gao. Lower and upper bounds for the blow-up time for nonlinear wave equation with variable sources. Computers and Mathematics with Applications, 71:267 – 277, 2016.
[41] S. Wu and L. Tsai. on global existence and blow up of solutions for an integro differential equation with strong damping. Taiwanese Journal of Mathematics, 10:979 – 1014, 2006.
[42] S. Wu and L. Tsai. Blow-up of positive-initial-energy solution for an integro-differential equation with nonlinear damping. Taiwanese Journal of Mathematics, 14:2043 – 2058, 2010.
